Problem 55 - Algebra 1: Concepts and Skills

Question

Simplify the expression. Use only positive exponents. $$ \frac{x^{2}}{x y^{-4}} \cdot \frac{2 x^{-3} y^{4}}{3 x y^{-1}} $$

Step-by-Step Solution

Answer

The simplified result is \( \frac{2}{3x^{3}}y^{9}\).

Step 1: Distribute the numerators and the denominators separately

First, distribute the numerators as \(x^{2} \cdot 2 x^{-3} y^{4} = 2x^{2-3}y^{4} = 2x^{-1}y^{4}\) and the denominators as \(x y^{-4} \cdot 3 x y^{-1} = 3x^{1+1}y^{-4-1} = 3x^{2}y^{-5}\). This makes use of the product of powers rule.

Step 2: Divide the results of Step 1 using the rules of exponents

Divide the result from the numerator by the result from the denominator, making use of the property that \(\frac{a^{n}}{a^{m}} = a^{n-m}\) and \(\frac{1}{a^{-n}} = a^{n}\). This gives: \( \frac{2x^{-1}y^{4}}{3x^{2}y^{-5}} = \frac{2}{3} \frac{x^{-1}}{x^{2}} \frac{y^{4}}{y^{-5}} = \frac{2}{3} x^{-1-2}y^{4-(-5)} = \frac{2}{3}x^{-3}y^{9}\).

Step 3: Convert to positive exponents

Lastly, for the term \(x^{-3}\) in our result, we convert it to positive exponent using \(a^{-n} = 1/a^n\). This gives us the final answer: \( \frac{2}{3}\cdot \frac{1}{x^{3}} \cdot y^{9} = \frac{2}{3x^{3}}y^{9}\).

Key Concepts

Simplifying Expressions

When you encounter complex expressions, it's important to simplify them. Simplifying expressions means manipulating them to their most basic or reduced form. This often involves rewriting expressions with positive exponents, combining like terms, or canceling out terms when possible.

In the given exercise, we dealt with fractions and multiple variables, which required us to separate the expressions into distinct parts and simplify them step by step. By working through each component separately—simplifying numerators and denominators independently—we can then restructure them to form the most reduced version of the expression.

Product of Powers Rule

The Product of Powers Rule is a fundamental concept in algebra that helps simplify expressions containing exponents. It states that when you multiply two powers with the same base, you can add the exponents: \( a^m \times a^n = a^{m+n} \).

In our exercise, we applied this rule to simplify both the numerators and the denominators. For instance, terms such as \( x^2 \cdot x^{-3} \) were simplified to \( x^{2-3} = x^{-1} \), significantly simplifying the overall expression. This rule is extremely useful when working with multiple terms raised to different powers and looking to condense them into a single term.

Positive Exponents

Working with positive exponents is desirable because they make expressions easier to understand and compare. A negative exponent indicates that the base is on the denominator side of a fraction. To convert a negative exponent to a positive one, you take the reciprocal of the base raised to the positive of that exponent: \( a^{-n} = \frac{1}{a^n} \).

In our final step, we transformed \( x^{-3} \) into \( \frac{1}{x^3} \) to ensure all the exponents in our final answer were positive. By doing so, we achieved a simplified expression, \( \frac{2}{3x^3}y^9 \), which is not only more standard but also more readable. Using positive exponents is a standard practice in mathematics as it aligns with the convention of presenting answers in their simplest form.